Problem: $g'(x)=-2g(x)$, and $g(0)=5$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $g(x)=e^{-2x}$ (Choice B) B $g(x)=5e^{-2x}$ (Choice C) C $g(x)=e^{-2x}+4$ (Choice D) D $g(x)=e^{-2x}+5$
Answer: The general solution of equations of the form $g'(x)=kg(x)$ is $g(x)=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=-2$, so $g(x)=C\cdot e^{-2x}$. Let's use the fact that $g(0)=5$ to find $C$ : $\begin{aligned} g(x)&=C\cdot e^{-2x} \\\\ g(0)&=C\cdot e^{-2\cdot 0} \gray{\text{Plug }x=0} \\\\ 5&=C\cdot e^{-2\cdot 0} \gray{g(0)=5} \\\\ 5&=C \end{aligned}$ In conclusion, $g(x)=5e^{-2x}$.